PoS(FFK2019)036

Study of the valence electronic density distributionin Z = 112−120 atoms

Mikhail Y. Kaygorodov∗

1 Department of Physics, St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St.Petersburg, RussiaE-mail: st031727@student.spbu.ru

Yury S. Kozhedub1 Department of Physics, St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St.Petersburg, RussiaE-mail: y.kozhedub@spbu.ru

Ilya I. Tupitsyn1 Department of Physics, St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St.Petersburg, Russia2 Center for Advanced Studies, Peter the Great St. Petersburg State Polytechnic University,Polytekhnicheskaya 29, St. Petersburg 195251, Russia3 St. Petersburg Nuclear Physics Institute, mkr. Orlova roshcha 1, Leningrad district, Gatchina,188300, RussiaE-mail: i.tupitsyn@spbu.ru

Vladimir M. Shabaev1 Department of Physics, St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St.Petersburg, RussiaE-mail: v.shabaev@spbu.ru

The localization properties of the valence electronic density are considered for superheavy atomsin the range of atomic numbers 112≤ Z ≤ 120, paying special attention to oganesson atom (Z =

118). As in [P. Jerabek et al., Phys. Rev. Lett. 111222000, (2018) 053001], the atomic shell structureis studied by means of the relativistic calculations using a non-relativistic expression for electronlocalization function. The influence of the relativistic effects on the distribution of the valenceelectronic density is evaluated by performing related calculations in the non-relativistic limit.One-electron Dirac-Fock and Hartree-Fock valence subshell densities are calculated and no signsof smearing out these densities are found.

International Conference on Precision Physics and Fundamental Physical Constants - FFK20199-14 June, 2019Tihany, Hungary

∗Speaker.

c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/

PoS(FFK2019)036

Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

1. Introduction

Superheavy elements (SHEs) are usually referred to artificially synthesized elements startingfrom rutherfordium (Z = 104) up to the last known element oganesson (Z = 118) which closes the8th row of the Periodic Table. It is a major challenge to study physical and chemical properties ofSHEs since the elements have short half-lives and low production rates. The lack of stable isotopesand demand on specific facilities are also a great issue experiment has to cope with. Therefore, themost available way to study physical and chemical properties of SHEs nowadays remains theoreti-cal calculations based on the most elaborated methods of relativistic quantum mechanics [1, 2].

Various theoretical calculations were performed to understand and predict physical propertiesof SHEs such as ionization potentials, energy levels, polarizabilities, electronic affinities [3, 4, 5,6, 7]. The calculations predict a growing impact of relativistic effects on the atomic shell structurewhat can lead to exotic and anomalous properties of SHEs compared with their lighter homologous.For example, in Ref. [3] it was shown that Og has a positive electron affinity, in spite of having theelectronic configuration of a noble gas. This means that Og may form a negatively charged ion.Another example of anomalous properties of SHEs is a recent study of Og electronic structure inRef. [8]. In this work it was concluded that due to relativistic effects the electron density becomesmore uniformly distributed over the entire atomic range and approaches the uniform-electron gasregime in the valence region. The conclusion was based on analysis of an electron localizationfunction (ELF), which was introduced first in Ref. [9] for the sake of visualization of the shellstructure in atoms and chemical bonding in molecules.

The present work is focused on examination of the results of Ref. [8] for Og atom and moredetailed study of the SHE electronic density in the valence region. The elements which have beenchosen to study are Og and its “neighbors” in the Periodic Table with the atomic charge numbers112 ≤ Z ≤ 120 and the electronic configurations [Rn]5 f 146d107sn7pm8sk, where 0 ≤ n ≤ 2, 0 ≤m≤ 6, 0≤ k ≤ 2. We examine the shell structure yielded by the ELF for both relativistic and non-relativistic one-electron densities obtained as the solutions of the single-configuration Dirac-Fockequations. The non-relativistic data are obtained by the non-relativistic limit (c→∞, c is the speedof light) of the Dirac-Fock calculations. The Dirac-Fock equations were solved using the Dirac-Fock-Sturm approach described in Ref. [10]. A non-relativistic method of the ELF evaluation,which incorporates some localized attraction potential, is also suggested. The approach allows usto obtain the ELF profile close to the relativistic one in the valence region. The relativistic and non-relativistic electron subshell density distribution widths, which can be considered as an indicatorof smearing out of the one-electron density, are studied as well.

The paper is organized as follows: in Sec. 2 the derivation of the non-relativistic expression forthe ELF is given, then the work is proceeded with numerical results which are presented in Sec. 3,and, finally Sec. 4 contains some conclusions and outlook.

2. Electron localization function

To study one-electron localization properties in the valence region one cannot rely on the totalone-electron density since strong overlapping of the shell densities makes impossible to revealthe shell structure. To extract the desired quantitative information about the shell structure one

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Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

could exploit the Laplacian, −∇2ρ , quantity −|∇ρ|/ρ and several other functions of the total one-electron density. The ELF is one of the instruments to perform qualitative analysis of atomic shellstructure in the 3D configuration space [9, 11, 12, 13].

In the ELF the electron localization is related to the smallness of the conditional pair proba-bility of finding a spin-like electron in vicinity of another one which in the Hartree-Fock approxi-mation is given by [9, 11]:

D(rrr) =12

[τ(rrr)− 1

4|∇ρ(rrr)|2

ρ(rrr)

], (2.1)

where the one-electron density ρ(rrr) and the kinetic energy density τ(rrr) are defined by

ρ(rrr) = ∑j|ϕ j(rrr)|2 , τ(rrr) = ∑

j|∇ϕ j(rrr)|2. (2.2)

Here the index j numerates the relativistic one-electron atomic wavefunctions ϕ j(rrr).The dimensionless localization index χ may be represented as a ratio χ = D(rrr)

D0(rrr), where D0(rrr)

corresponds to the case, when the uniform electron gas relation between the particle density and thekinetic-energy density is considered for bound electrons in the atomic potential under consideration(Thomas-Fermi atomic model),

D0(rrr) =310

(3π)2/3ρ(rrr)5/3. (2.3)

With these conventions we define the ELF (η(rrr)) as

η(rrr) =1

1+χ2(rrr). (2.4)

We emphasize that for atomic configurations with closed shells ρ(rrr), τ(rrr) and, therefore, D(rrr)and D0(rrr) depend only on r = |rrr|. For configurations with open shells it is not true. However, onecan average ρ(rrr) and τ(rrr) over angles and then consider D(rrr) and D0(rrr) again as functions of r.

The ELF is normalized in such a way that the low electronic density stands for values of theELF close to 0 and the regions of strong electron localization correspond to values of the ELF closeto 1. It should be noted that for the uniform electron gas ρ is constant, the ELF η = 0.5.

3. Results

To confirm that our definition of the ELF agrees with one used in Ref. [8], in Fig. 1a andFig. 1b we present a comparison of the ELF plot for Og atom obtained in the present work withthe ELF plot which was taken from Ref. [8]. Both figures display the ELF for Og atom for therelativistic and non-relativistic calculations. One can see perfect agreement between these tworesults. For the relativistic calculations the ELF in the valence region is indeed close to 0.5, but inthe non-relativistic case a pronounced shell structure of Og can be observed. However, we warnthe reader that in [8] the x-axis entry should be ln(r/a0) instead of log10(r/a0).

Let us now examine the Dirac-Fock and Hartree-Fock electron subshell density distributionwidths. We define the widths as the standard deviation (STD) of the corresponding subshell densitydistribution. It is expected that if there is some sort of smearing out of the atomic shell structure

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PoS(FFK2019)036

Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

-4 -3 -2 -1 0 1 2

ln(r/a0)

0.0

0.2

0.4

0.6

0.8

1.0

η(r

)

NRR

n=7

n=6n=5n=4

n=3n=2

n=1

(a) Present work.

1.0

0.0

0.2

0.4

0.6

0.8

-4 -3 -2 -1 0 1 2

NR R

(b) Plot from Jerabek et al. [8].

Figure 1: ELFs for Og atom from the non-relativistic (NR) and relativistic (R) calculations asfunctions of the distance from the nucleus.

then the one-electron subshell density should be smeared out as well. For elements with the atomiccharge numbers 112 ≤ Z ≤ 120, in Fig. 2 we plot STD of the valence subshell electronic densitydistributions. However, as one can see from Fig. 2 the widths of the Dirac-Fock and non-relativisticHartree-Fock subshells tend to be sharper with increasing atomic number. Therefore, the relativisticand non-relativistic one-particle subshell widths do not show any behavior similar to the uniformelectron gas. The break in the smooth decrease of the STDs after Z = 118 is attributed to appearanceof electrons on the 8s orbital.

Figure 2: The standard deviations (STDs) of the valence electron subshell densities of the SHEseries. The dashed lines stand for the Dirac-Fock subshell widths and the dotted lines indicate theHartree-Fock ones.

The ELF for the sequence of atoms with the atomic charge numbers 112≤ Z ≤ 120 is studiedto comprehend whether the function predicts smearing out of the valence shell structure only forOg atom or for the other atoms as well. Fig. 3 shows the ELF in the valence region for atoms with112≤ Z ≤ 120 in the non-relativistic calculations. The green dots in the Z− ln(r/a0) plane indicatethe root-mean square (RMS) radii of the Hartree-Fock subshell electron density distribution. We

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Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

note that in the valence region the maxima of the ELF do not coincide with the positions of theRMS radii. The distinct ELF peaks correspond to the n = 6 and n = 7 shells in the non-relativisticcalculations. The known artifact of the ELF for atoms with only valence ns orbitals occupied [9] iseliminated in the figure.

Figure 3: The ELFs for the valence region in the non-relativistic calculations for 112 ≤ Z ≤ 120elements.

The results of the relativistic calculations are presented in Fig. 4. These calculations of the ELFalong the considered SHE sequence show that Og atom has the most pronounced shell structurein the valence region. The incongruity of the RMS radii of the Dirac-Fock subshells and the ELFpeaks is observed in the relativistic calculations as well. The relativistic calculations give valuesof the ELF close to 1 for atoms with Z = 119 and Z = 120, where the 8s orbital is occupied.However, the ELF for the Z = 112 element, where the 7s orbital is fully occupied, does not showthis behavior.

To investigate whether the behavior of the ELF in the relativistic calculations is indeed due tostrong relativistic effects we simulate the contraction of the valence s and p orbitals by incorpo-rating a highly localized attraction potential into the calculations of the 7s and 7p1/2 states withinthe non-relativistic limit (c→ ∞) of the Dirac-Fock equations. In these calculations all electronsbecome non-relativistic, and the 7s and 7p1/2 states are shifted closer to the nucleus. The attractionpotential was taken in the form (in a.u.)

VAtt.Pot.(r) =−22500e−140r. (3.1)

This form is chosen to get the simplest strongly localized potential for the states under consid-eration. Fig. 5 demonstrates how the described incorporation of VAtt.Pot. into the non-relativisticcalculations changes the shape of the ELF for Og atom. In the plot the relativistic (blue dashedline), non-relativistic (black solid line) and non-relativistic with VAtt.Pot. (orange dotted line) ELFsare shown. The profile of the ELF in the non-relativistic calculations with VAtt.Pot. in the valenceregion is close to the profile of the ELF in the relativistic calculations. This means that only the

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Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

Figure 4: The ELFs for the valence region in the relativistic calculations for 112 ≤ Z ≤ 120 ele-ments.

contraction of the 7s and 7p1/2 orbitals, which leads to the shift of the 7s and 7p1/2 orbitals closerto the main maxima of the outer core 6d orbital, is responsible for the fact that the ELF is close to0.5 in the valence region.

Although all the results presented have been obtained within the framework of the single-configuration Dirac-Fock method, we also have studied the role of the configuration-interaction(CI) effects on the smearing out of the valence shell densities in Og atom. It is known that theseeffects can be very important in calculations of various properties of SHEs [14]. The CI-DF calcula-tions of the electronic structure of Og show, however, that the CI weight of the ground configuration[Rn]5 f 146d107s27p6 is about 94%. This means that other configurations do not give any noticeablecontributions despite the supposed "smearing out" of the valence shells.

4. Conclusion and Outlook

The present work was aimed at studying the electron localization properties of the valenceelectron density for SHEs with the atomic charge numbers 112≤ Z ≤ 120. We investigated in de-tails the results based on the ELF expression [9] for these elements. It was found that the relativisticcalculations predict smearing out of the valence electronic density for all SHEs under considera-tion, while the non-relativistic calculations yield a distinct structure of atomic orbitals. In contrastto the ELF results, the calculations of the one-electron Dirac-Fock and non-relativistic Hartree-Fock electron subshell density widths do not indicate any signs of smearing out of the electrondensity distribution. The widths remain finite and with increasing the atomic number they becomeeven smaller. If there were a case of delocalization in the valence region, then the valence one-electron density would be smeared out as well. To study the origin of smearing out of the ELF inthe relativistic calculations, we have also performed the following test. In the non-relativistic cal-culations, where the ELF predicts a sharp shell structure, we artificially inserted a local attractionpotential into the equations for the valence s and p orbitals. This procedure simulates the relativis-

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PoS(FFK2019)036

Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

-4 -3 -2 -1 0 1 2

ln(r/a0)

0.0

0.2

0.4

0.6

0.8

1.0

η(r

)

NRRNR+V

Att.Pot.

n=7

n=6n=5n=4

n=3n=2

n=1

Figure 5: The ELFs in the relativistic (R), non-relativistic (NR) and non-relativistic with attractionpotential (NR+VAtt.Pot.) calculations for Og atom.

tic contraction of the considered orbitals. The resulted ELF in the valence region has almost thesame profile as the ELF in the relativistic calculations. Based on these results, we conclude thatthe smearing out of the ELF is due to the shifts of the 7s and 7p1/2 orbitals towards the 6d orbitalcaused by the relativistic effects.

5. Acknowledgments

This work is supported by the Russian Foundation for Basic Research (RFBR) (Grants No. 18-03-01220, 17-02-00216) and by SPbSU-DFG (Grants No. 11.65.41.2017, No. STO 346/5-1). M.Y.K.and V.M.S. acknowledge the support of the foundation for the advancement of the theoreticalphysics and mathematics “Basis”. I.I.T. thanks the Ministry of Education and Science of the Rus-sian Federation (Grant No. 3.1463.2017/4.6) and St. Petersburg State University (Grant No. 37717909).

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Study of the valence electronic density in SHEs Mikhail Y. Kaygorodov

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